Finite Group Databasebeta

← #13159347

Subgroups — #13159347

Order 1728.

888 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#895C211yesmaximal, characteristic
2#11723426C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic, complemented
4C2×#88786C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6C3×#63634C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592741—31nomaximal
9#10590588—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×D12—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19#88807—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88794—22no—
22#88786—22no—
23C32×#121—31no—
24C2×#3754—31no—
25#64133—31no—
26C2×#3748—31no—
27C3×#895D314yes—
28#63857—34no—
29C2×#3716—34no—
30C2×#3711—34no—
31#63659—31no—
32#63637—31no—
33#63637—34no—
34#63637—64no—
35#63634D311yescharacteristic, complemented
36C3×#807—34no—
37#63562—34no—
38C22×C32×D3—21no—
39C22×C32×D3D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C32×D12—21no—
48C32×D12—21no—
49C32×D12—42no—
50C32×C4×D3—21no—
51C32×C4×D3—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54#5042—94no—
55#4886—91no—
56C2×C32×D4—31no—
57C2×C32×D4—61no—
58C22×C32×C4—31no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×D12—24no—
63C2×C3×D12C2×D314yes—
64C2×C3×C4×D3C2×D314yes—
65C32×C4⋊C4—24no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C22⋊C4—61no—
69#3757—61no—
70#3754—62no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72#3750—61no—
73#3748—62no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×(C2×D3)⋊C4—24no—
79C3×C4⋊Dic3—34no—
80C2×C3×C3⋊C8—34no—
81Dic3⋊Dic3—64no—
82(C2×D3)⋊Dic3—64no—
83#3716—68no—
84#3711—68no—
85C22×C33C2×D411yescharacteristic
86C2×C32×D3—21no—
87C2×C32×D3—21no—
88C2×C32×D3—21no—
89C2×C32×D3—41no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—21no—
98C3×#121—34no—
99C2×#214—94no—
100#924—94no—
101C2×#208—94no—
102C2×#204—91no—
103C2×ΓL(1,𝔽25)—91no—
104#897—91no—
105#895C3⋊D311yescharacteristic
106C3×#107—91no—
107#832—94no—
108#810—94no—
109#810—91no—
110#810—184no—
111#809—91no—
112#807—34no—
113C23×C32—31no—
114C23×C32—61no—
115C22×C3×D3—24no—
116C22×C3×D3#6714yes—
117C32×D4—61no—
118C32×D4—61no—
119C32×D4—122no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—31no—
124C2×C32×C4—44no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—124no—
133#634—68no—
134#634—61no—
135#634—64no—
136#634—61no—
137#634—64no—
138#634—124no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3#6714yes—
141C3×D12—24no—
142C3×D12—24no—
143C3×D12—48no—
144C3×C4×D3—24no—
145C3×C4×D3—24no—
146C3⋊(C3⋊C8)—61no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—124no—
149C3⋊(C3⋊C8)D1212yescomplemented
150C3×C3⋊C8—38no—
151#468—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C2×C33#13711yescharacteristic
155C32×D3—41no—
156C32×D3—81no—
157C2×C3×D4—34no—
158C2×C3×D4—64no—
159C22×C3×C4—34no—
160C2×D12—21no—
161C2×D12C2×C3⋊D311yescharacteristic
162C2×C4×D3C2×C3⋊D311yescharacteristic
163C2×Dic6—94no—
164C2×Dic6—91no—
165C2×Dic6—184no—
166C2×C3×C8—91no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4D3214yes—
170C3×C4⋊C4C2×C3⋊D311yescharacteristic
171C3×C22⋊C4—64no—
172#217—184no—
173#214—188no—
174(C2×D3)⋊C4—21no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—34no—
177C4⋊Dic3—181no—
178Dic3⋊C4—184no—
179#208—188no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8—94no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184#204—182no—
185ΓL(1,𝔽25)—182no—
186C22×C32—24no—
187C22×C32—61no—
188C22×C32—61no—
189C22×C32—61no—
190C22×C32—121no—
191C22×C32—121no—
192C22×C32—121no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×#6714yes—
195C2×C3×D3—24no—
196C2×C3×D3—24no—
197C2×C3×D3—24no—
198C2×C3×D3—44no—
199C2×C3×D3—44no—
200C2×C3×D3—44no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—41no—
204C32×C4—44no—
205C32×C4—61no—
206C32×C4—84no—
207C32×C4C2×D1211yescharacteristic
208C32×C4D3×D414yes—
209C32×C4#23714yes—
210C32×C4#23911yescharacteristic
211C3⋊Dic3—41no—
212C3⋊Dic3—128no—
213C3⋊Dic3—121no—
214C3⋊Dic3—124no—
215C3×Dic3—24no—
216C3×Dic3—124no—
217C2×SD16—271no—
218C2×M16—271no—
219C22⋊Q8—271no—
220#121—31no—
221#107—91no—
222#103—271no—
223#102—271no—
224C33#46811yescharacteristic, complemented, Sylow
225C23×C3—34no—
226C23×C3—64no—
227C22×D3—21no—
228C22×D3#63811yescharacteristic
229C3×D4—64no—
230C3×D4—64no—
231C3×D4—128no—
232C2×C3×C4—24no—
233C2×C3×C4—21no—
234C2×C3×C4—24no—
235C2×C3×C4—34no—
236C2×C3×C4—44no—
237C2×C3×C4—64no—
238C2×C3×C4—64no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242C2×Dic3—64no—
243C2×Dic3—184no—
244C2×Dic3—181no—
245C2×Dic3—364no—
246C2×Dic3#63811yescharacteristic
247D12—21no—
248D12—21no—
249D12—42no—
250C4×D3—21no—
251C4×D3—21no—
252Dic6—188no—
253Dic6—184no—
254Dic6—184no—
255Dic6—181no—
256Dic6—181no—
257Dic6—364no—
258C3×C8—92no—
259C3⋊C8—38no—
260C3⋊C8—184no—
261C3⋊C8—181no—
262C3⋊C8—364no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—121no—
267C2×C32—241no—
268C2×C32#89511yescharacteristic
269C2×C32#91111yescharacteristic
270C2×C32#91811yescharacteristic
271C2×C32#93214yes—
272C2×C32#93914yes—
273C2×C32#95114yes—
274C3×D3—44no—
275C3×D3—84no—
276C2×Q8—271no—
277C2×D4—31no—
278C2×D4—61no—
279C22×C4—31no—
280SD16—542no—
281M16—542no—
282C2×C8—91no—
283C2×C8—271no—
284C4⋊C4—91no—
285C4⋊C4—541no—
286C4⋊C4D3×C3⋊D311yescharacteristic
287C22⋊C4—61no—
288C22⋊C4—541no—
289C22×C3—24no—
290C22×C3—64no—
291C22×C3—64no—
292C22×C3—64no—
293C22×C3—124no—
294C22×C3—124no—
295C22×C3—124no—
296C22×C3D3×#6714yes—
297C22×C3C2×#63811yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—41no—
307C3×C4—44no—
308C3×C4—64no—
309C3×C4—84no—
310C3×C4—361no—
311C3×C4#379614yes—
312C3×C4D3×D1214yes—
313C3×C4D4×C3⋊D311yescharacteristic
314C3×C4#383011yescharacteristic
315Dic3—21no—
316Dic3—124no—
317Dic3—368no—
318Dic3—364no—
319Dic3—361no—
320C32—24no—
321C32#488611yescharacteristic, complemented
322C32#504214yescomplemented
323C23—31no—
324C23—61no—
325Q8—541no—
326Q8—541no—
327D4—61no—
328D4—61no—
329D4—122no—
330C2×C4—21no—
331C2×C4—31no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—181no—
335C2×C4—541no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—92no—
338C8—541no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—124no—
343C2×C3—244no—
344C2×C3#6388614yes—
345C2×C3#6393214yes—
346C2×C3#6394314yes—
347C2×C3#6414111yescharacteristic
348C2×C3#6414811yescharacteristic
349C2×C3#6416011yescharacteristic
350D3—41no—
351D3—81no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22D3×#63811yescharacteristic
359C4—41no—
360C4—61no—
361C4—361no—
362C4—1081no—
363C4#8901511yescharacteristic
364C4C3⋊D3×D1211yescharacteristic
365C3—24no—
366C3#1059058814yescomplemented
367C3#1059274111yescharacteristic, complemented
368C2—121no—
369C2—241no—
370C2#1172347511yescharacteristic
371C2#1172355511yescharacteristic
372C2#1172357411yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC32×#895C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723426C2semidirect
3. C2×#88794C2semidirect
4. C2×#88786C2semidirect
5. #11722180C2semidirect
6. C3×#63634C2semidirect
7. #11722031C2semidirect
35. #63634D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
149. C3⋊(C3⋊C8)D12semidirect
224. C33#468semidirect
321. C32#4886semidirect
322. C32#5042semidirect
366. C3#10590588semidirect
367. C3#10592741semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#8952
#117234262
C2×#887942
C2×#887862
#117221802
C3×#636342
#117220312
#105927413
#105905883